16.06 Principles of Automatic Control
Lecture 29
Digital Control
At one time, most control systems were implemented using analog devices (operational am­
plifiers, linear circuit elements, etc). Today, most control systems are implemented using
digital devices (computers, etc.)
The use of digital devices leads to systems which implement difference equations in discrete
time rather than differential equations in continuous time. This will require some new tools,
but much will be familiar.
Basic block diagram:
sampler
+ error
reference
r(t)
+
T
controller
- e(kt)
D/A + hold
u(kt)
G(s)
u(t)
y(t)
A/D
T
Discrete-time components
Basic Components:
Sampler - Captures value of continuous time signal periodically so that A/D converter can
read it
Analog-to-digital (A/D) converter - converts samples continuous signal to discrete (bi­
nary encoded) signal. Often, 8, 12, or 14 bits.
1
Controller - Implemented on a computer, which implements a difference equation, much as
a continuous-time controller is an implementation of a differential equation on analog
components.
Digital-to-analog (D/A) converter - Converts a binary word to an analog signal.
Zero Order Hold - Holds a constant value analog signal for one period.
Discrete Time Control Time line
Samples values
of r, y converted
to digital values
Digital value of u
converted to
analog signal
Control Computer
calculates next
value of u
kT
Control Computer
performs other
tasks
kT+δ
(k+1)T
If the control computer is dedicated to a single task, then T is made as short as possible,
and δ “ T.
If the computer has many tasks, then T is longer, and δ ! T . In that case, can take δ “ 0.
Of course, can have anything in between.
For our purposes, we will assume one of the extremes, i.e., either δ “ 0, or δ “ T. Of course,
can generalize to other cases.
Analysis of Discrete Systems.
Our system is a sampled-data system, meaning that it contains both discrete and continuous
signals. However, we can treat it as a discrete system, by treating the dynamics from vpkT q
to ypkT q as discrete.
Discrete systems are a lot like continuous systems - they have step responses, pulse (not
impulse) responses, and transfer functions. The counterpart of the Laplace Transform is the
Z-Transform.
Z-Transform
The Z-transform of a sequence f rks is given by
2
F pzq “ Z rf rkss “
ÿ
8
´k
´8 f rksz
Note: I am using the bilateral transform.
For every interesting property that you learned for the Laplace transforms, there is a corre­
sponding z-transform property.
For instance,
Z rf rk ´ 1ss “
8
ÿ
f rk ´ 1sz ´k
k“´8
8
ÿ
“
1
f rk 1 sz ´pk `1q
k1 “´8
´1
“z F pzq
(similar to Lrf9ptqs “ sF psq )
Using this property, can always differentiate equations. E.g.,
yrks “ ´a1 yrk ´ 1s ´ a2 yrk ´ 2s ` b0 urks ` b1 urk ´ 1s ` b2 urk ´ 2s
Z-transform both sized to obtain
Y psq “ p´a1 z ´1 ´ a2 z ´2 qY pzq ` pb0 ` b1 z ´1 ` b2 z ´2 qU pzq
So the transfer function from u to y is
Y pzq b0 ` b1 z ´1 ` b2 z ´2
“
U pzq 1 ` a1 z ´1 ` b2 z ´2
b0 z 2 ` b1 z ` b2
“ 2
z ` a1 z ` a2
Some Select Transforms
3
#
1,
k“0
δrks “
0,
1,
ô
z ´k0
ô
z
z´1 ,
ô
z
z´α1 ,
ô
z
z´e´aT
kě0
σrks “
0,
|z| ą 1
kă0
σrksαk
σrkse´akT
1
k‰0
δrk ´ k0 s
#
ô
`
1
s`a
˘
4
|z| ą |α|
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16.06 Principles of Automatic Control
Fall 2012
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